Question: Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be positive real numbers such that $a + b + c + d + e + f = 7.$  Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d} + \frac{25}{e} + \frac{36}{f}.\]
Explanation: By Cauchy-Schwarz,
\[(a + b + c + d + e + f) \left( \frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d} + \frac{25}{e} + \frac{36}{f} \right) \ge (1 + 2 + 3 + 4 + 5 + 6)^2 = 441,\]so
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d} + \frac{25}{e} + \frac{36}{f} \ge \frac{441}{7} = 63.\]Equality occurs when $a^2 = \frac{b^2}{4} = \frac{c^2}{9} = \frac{d^2}{16} = \frac{e^2}{25} = \frac{f^2}{36}$ and $a + b + c + d + e + f = 7.$  Solving, we find $a = \frac{1}{3},$ $b = \frac{2}{3},$ $c = 1,$ $d = \frac{4}{3},$ $e = \frac{5}{3},$ and $f = 2,$ so the minimum value is $\boxed{63}.$